The above stated makes meaningful to define the norm and the trace functions in an algebraic number field as follows.

Definition. If αα\alpha is an element of the number field ℚ⁢(ϑ)ℚϑ\mathbb{Q}(\vartheta), then the normN⁢(α)Nα\mbox{N}(\alpha) and the traceS⁢(α)Sα\mbox{S}(\alpha) of αα\alpha are the product and the sum, respectively, of all ℚ⁢(ϑ)ℚϑ\mathbb{Q}(\vartheta)-conjugatesα(i)superscriptαi\alpha^{{(i)}} of αα\alpha.

Since the coefficients of the characteristic equation of αα\alpha are rational, one has